# Automorphisms and derivations in affine algebraic geometry

Mini-course by

## Professor Leonid Makar-Limanov

Wayne University

USA

March 13 - 17, 2023

Professor Makar-Limanov (read the mini course poster here and see more pictures below)

### (See Lecture 1 and Lecture 2; for Lecture 3 see Journal of Algebra and Applications, Vol. 14, No. 9 (2015) 1540001)

Brief description of the mini course

After this course you will know the proofs of several classical theorems of Affine Algebraic Geometry. The original proofs of these theorems were quite involved and a much longer course would be needed for their exposition.

In the first lecture we will discuss the theorems of Heinrich Jung and Rudolf Rentschler. The first one describes all invertible transformations of the plane by polynomials and the second all generalized shifts of the plane. Algebraically speaking, Jung's theorem describes all automorphisms of the ring of polynomials with two variables and Rentschler theorem describes all subgroups of this group which are isomorphic to the group of complex numbers under addition. If we have time, we will  discuss the groups of polynomial automorphisms of several other surfaces.

The second lecture is devoted to the following topic: if a cylinder is given, is it possible to recover the base of this cylinder. In general the answer is no, but we discuss two cases when this is possible. We show that if the cylinder over a curve is given then we can recover this curve (this is the theorem of Shreeram Abhyankar, Paul Eakin, and William Heinzer). If the cylinder over a surface is isomorphic to a three-dimensional space then the surface is isomorphic to a plane (this is a theorem of Takao Fujita).

Here is an algebraic translation:

If A is an integral domain of transcendence degree one and A[x1, x2,…, xn] is given, we can recover A up to an isomorphism. If A is an integral domain of transcendence degree two and A[x] is isomorphic to C[y1,y2,y3] then A is isomorphic to C[z1,z2].  The main tool used in these two lectures is locally nilpotent derivations.

In the third lecture we prove one of the most famous theorems in affine algebraic geometry, the AMS Theorem (after Abhyankar, Tsuong-tsieng Moh, Masakazu Suzuki): any smooth "good" embedding of a line to a plane is the image of a coordinate line under an automorphism of the plane. Algebraically, this means the following: if two polynomials f(t), g(t)C[t] generate C[t] then the smaller of the degrees of f(t), g(t) divides the larger of the degrees of f(t), g(t). The main tool here is a new algorithm for finding an irreducible dependence between two polynomials in one variable.

The lectures will be delivered during three time periods, as shown below. They will take place at the St. John's campus of Memorial University and will be broadcast via Webex. All the times are in Newfoundland Time (NST=UTC-3:30).

Monday March 13 room A2065  from 3:30-5pm

Wednesday March 15 room A2065  from 3:30-5pm

Thursday March 16th room ED4015  from 11am-12:30pm.

The lectures will be available online via Webex.  The details are as follows.

On Monday and Wednesday:

 More ways to join:
 Join from the meeting link https://mun.webex.com/mun/j.php?MTID=m93ec262405e6d7e8e05482bd7e52cba8
 Join by meeting number Meeting number (access code): 2772 379 3939 Meeting password: bmR6UUyi58r
 Tap to join from a mobile device (attendees only) +1-709-722-5626,,27723793939## Canada Toll (St.Johns) +1-416-915-6530,,27723793939## Canada Toll
 Join by phone +1-709-722-5626 Canada Toll (St.Johns) +1-416-915-6530 Canada Toll Global call-in numbers
 Join from a video system or application Dial 27723793939@mun.webex.com You can also dial 173.243.2.68 and enter your meeting number.

On Thursday:

 More ways to join:
 Join from the meeting link https://mun.webex.com/mun/j.php?MTID=m177a598f0789f6c3b6dc2bff46d533f7
 Join by meeting number Meeting number (access code): 2770 308 4266 Meeting password: WAm3piJ5cM5
 Tap to join from a mobile device (attendees only) +1-709-722-5626,,27703084266## Canada Toll (St.Johns) +1-416-915-6530,,27703084266## Canada Toll
 Join by phone +1-709-722-5626 Canada Toll (St.Johns) +1-416-915-6530 Canada Toll Global call-in numbers
 Join from a video system or application Dial 27703084266@mun.webex.com You can also dial 173.243.2.68 and enter your meeting number.